,

$$\begin{eqnarray*} A & = & \left(0.5I_{3}\right)^{-1}\\  \det A & = & \det\left[\left(0.5I_{3}\right)^{-1}\right]\\ & = & \frac{1}{\det\left(0.5I_{3}\right)}\\ & = & \frac{1}{\left(0.5\right)^{3}\det I}\\ & = & \frac{1}{\left(0.5\right)^{3}\left(1\right)}\\ & = & 8 \end{eqnarray*}$$

,

$$\det{\operatorname{adj}A}=\left(\det A\right)^{n-1}$$

$$\begin{eqnarray*} \det B & = & \det\left(\operatorname{adj}A\right)\\ & = & \left(\det A\right)^{3-1}\\ & = & \left(\det A\right)^{2} \end{eqnarray*}$$

,

$$\begin{eqnarray*} \det\left(\operatorname{adj}\left(\operatorname{adj}A\right)\right) & = & \det\left(\operatorname{adj}B\right)\\ & = & \left(\det B\right)^{3-1}\\ & = & \left[\det B\right]^{2}\\ & = & \left[\left(\det A\right)^{2}\right]^{2}\\ & = & \left[\det A\right]^{4}\\ & = & 8^4 \end{eqnarray*}$$